arXiv subjects

G. M. Viswanathan

Publications and source records attributed to G. M. Viswanathan.

At least 19 recordsLinked to original sources

arXiv2026

Lévy walkers inside spherical shells with absorbing boundaries: Towards settling the optimal Lévy walk strategy for random searches

The Lévy flight foraging hypothesis states that organisms must have evolved adaptations to exploit Lévy walk search strategies. Indeed, it is widely accepted that inverse square Lévy walks optimize the search efficiency in foraging with unrestricted revisits (also known as non-destructive foraging). However, a mathematically rigorous demonstration of this for dimensions $D \geq 2$ is still lacking. Here we study the very closely related problem of a Lévy walker inside annuli or spherical shells with absorbing boundaries. In the limit that corresponds to the foraging with unrestricted revisits, we show that inverse square Lévy walks optimize the search. This constitutes the strongest formal result to date supporting the optimality of inverse square Lévy walks search strategies.

cond-mat.stat-mech↗

arXiv2025

How Quorum Sensing Shapes Clustering in Active Matter

Self-propelled particles undergoing persistent motion can accumulate either through excluded-volume interactions or through quorum sensing, where self-propulsion decreases at high local density. Using kinetic balance theory and simulations, we show that the interplay of these two mechanisms produces a reentrant, non-monotonic behavior in which clustering passes through a pronounced minimum as quorum-sensing strength or persistence time varies. Beyond a threshold quorum-sensing strength, we find long-lived transient states that retain memory of initial conditions, including kinetically arrested active gels. Although quorum sensing can mimic attractive interactions, it also acts strongly in dilute regions, producing an effective cluster bistability that is captured by our theory. Our results explain collective states observed experimentally in synthetic and biological active systems.

cond-mat.soft↗

arXiv2025

Relativistic Lévy processes

We study sums of independent and identically distributed random velocities in special relativity. We show that the resulting one-dimensional velocity distributions are not only stable under relativistic velocity addition but define a genuinely new class of stochastic processes--relativistic Lévy processes. Given a system, this allows identifying distinct relativistic regimes in terms of the distribution's concavity at the origin and the probability of measuring relativistic velocities. These features provide a protocol to assess the relevance of stochastic relativistic effects in actual experiments. As supporting evidence, we find agreement with previous results about heavy-ion diffusion and show that our findings are consistent with the distribution of momentum deviations observed in measurements of antiproton cooling.

cond-mat.stat-mech↗

arXiv2025

Intermediate statistics: addressing the Landau diamagnetism problem

Quantum groups and quantum algebras have received considerable attention in the last decades because they are very useful as mathematical tools of research. Existing proposals for quantum groups have always suggested the idea of deforming a classical object. Motivated by the possibility of anyons in three dimensions ($d=3$), with important consequences to a wide range of fields of physics, in the present work we investigate how the magnetization and other thermodynamic quantities, associated to the Landau diamagnetism problem, depend on the deforming parameter of two models with intermediate statistics: (i) $q$-fermions and (ii) $F$-anyons, and make {\it comparisons between both cases}. In particular, we extend the results from the literature for $q$-fermions by considering {\it second order terms} in the expansion of the grand partition function. Also, we find that for $F$-anyons statistics the magnetization shows a stronger response with respect to magnetic fields compared to magnetization for $q$-fermions statistics. This theoretical outcome may be experimentally verified for instance in superconductors, that are perfect diamagnetic materials with strong magnetic susceptibility, by adjusting impurities or pressure. The latter can be associated to the deforming parameter $q$.

cond-mat.stat-mech↗

arXiv2024

Intermediate statistics: addressing the thermoelectric properties of solids

We study the thermodynamics of a crystalline solid by applying intermediate statistics obtained by deforming known solid state models using the mathematics of $q$-analogs. We apply the resulting $q$-deformation to both the Einstein and Debye models and study the deformed thermal and electrical conductivities and the deformed Debye specific heat. We find that the $q$-deformation acts in two different ways -- but not necessarily as independent mechanisms. First, it acts as an effective factor of disorder or impurity, modifying the characteristics of a crystalline structure, which are phenomena described by q-bosons. Second, it also manifests intermediate statistics, namely, the B-anyons (or B-type systems). For the latter case, we have identified the Schottky effect, normally associated with high-$T_c$ superconductors in the presence of rare-earth-ion impurities. We also find that it increases the specific heat of the solids beyond the Dulong-Petit limit at high temperature. Such an effect is usually related to anharmonicity of interatomic interactions. Alternatively, since in the $q$-bosons case the statistics are in principle maintained, the effect of the deformation acts more slowly due to a small change in the crystal lattice. On the other hand, B-anyons that belong to modified statistics are more sensitive to the deformation. The results reported here may be verified experimentally, for instance, in experimental samples by inserting impurities, or changes in pressure or temperature if one assumes these tuning quantities are related with the q-deformation parameter.

cond-mat.stat-mech↗

arXiv2023

A fermionic path integral for exact enumeration of polygons on the simple cubic lattice

Enumerating polygons on regular lattices is a classic problem in rigorous statistical mechanics. The goal of enumerating polygons on the square lattice via fermionic path integration was achieved using a free-fermion quadratic action in the late 1970s. Given that polygon edges only link 2 vertices, it is considered plausible, if not natural, that an action of degree 2 in the Grassmann variables might suffice to enumerate lattice polygons in any dimension. Nevertheless, on nonplanar lattices the problem has remained open for more than four decades. Here we derive the Grassmann action for exact enumeration of polygons on the simple cubic lattice. Moreover, we prove that this action is not quadratic but quartic -- corresponding to a model of interacting fermions.

cond-mat.stat-mech↗

arXiv2023

Threshold-free estimation of entropy from a Pearson matrix

There is demand in diverse fields for a reliable method of estimating the entropy associated with correlations. The estimation of a unique entropy directly from the Pearson correlation matrix has remained an open problem for more than half a century. All existing approaches lack generality insofar as they require thresholding choices that arbitrarily remove possibly important information. Here we propose an objective procedure for directly estimating a unique entropy of a general Pearson matrix. We show that upon rescaling the Pearson matrix satisfies all necessary conditions for an analog of the von Neumann entropy to be well defined. No thresholding is required. We demonstrate the method by estimating the entropy from neuroimaging time series of the human brain under the influence of a psychedelic.

physics.data-an↗

arXiv2022

What does it take to solve the 3D Ising model? Minimal necessary conditions for a valid solution

Exact solution of the Ising model on the simple cubic lattice is one of the long-standing open problems in rigorous statistical mechanics. Indeed, it is generally believed that settling it would constitute a methodological breakthrough, fomenting great prospects for further application, similarly to what happened when Lars Onsager solved the two dimensional model eighty years ago. Hence, there have been many attempts to find analytic expressions for the exact partition function $Z$, but all such attempts have failed due to unavoidable conceptual or mathematical obstructions. Given the importance of this simple yet paradigmatic model, here we set out clear-cut criteria for any claimed exact expression for $Z$ to be minimally plausible. Specifically, we present six necessary -- but not sufficient -- conditions that $Z$ must satisfy. These criteria will allow very quick plausibility checks of future claims. As illustrative examples, we discuss previous mistaken ``solutions,'' unveiling their shortcomings.

cond-mat.stat-mech↗

arXiv2022

Spectrum of the tight-binding model on Cayley Trees and comparison with Bethe Lattices

There are few exactly solvable lattice models and even fewer solvable quantum lattice models. Here we address the problem of finding the spectrum of the tight-binding model (equivalently, the spectrum of the adjacency matrix) on Cayley trees. Recent approaches to the problem have relied on the similarity between Cayley tree and the Bethe lattice. Here, we avoid to make any ansatz related to the Bethe lattice due to fundamental differences between the two lattices that persist even when taking the thermodynamic limit. Instead, we show that one can use a recursive procedure that starts from the boundary and then use the canonical basis to derive the complete spectrum of the tight-binding model on Cayley Trees. Our resulting algorithm is extremely efficient, as witnessed with remarkable large trees having hundred of shells. We also shows that, in the thermodynamic limit, the density of states is dramatically different from that of the Bethe lattice.

cond-mat.stat-mech↗

arXiv2022

Numerical studies for an ab initio investigation into the Boltzmann prescription in statistical mechanics of large systems

We present numerical investigations into the question of the validity of the Boltzmann prescription in Statistical Mechanics for large systems, addressing the issue of whether extensivity of energy implies the extensivity of the Boltzmann entropy. The importance of the question stems from the fact that it is currently considered open by some investigators but quite settled by others. We report ab initio results for gas-like Hamiltonian systems with long-range as well as short-range interactions, based on simulations that explicitly consider more than $2^{30} \approx 10^9$ states of the full Hilbert space. The basis of the technique is Monte Carlo algorithms. Despite the largeness of the numbers used, careful inspection shows that the systems studied are still too small to settle uniquely the issues raised. Therefore, the new approach outlined represents a first step in addressing on first principles the question of non-extensive statistical mechanics. General theoretical comments are also supplied to supplement the numerical investigations.

cond-mat.stat-mech↗

arXiv2021

Comment on "Inverse Square Lévy Walks are not Optimal Search Strategies for d $\geq$ 2" [Phys. Rev. Lett. 124, 080601 (2020)]

It is widely accepted that inverse square Lévy walks are optimal search strategies because they maximize the encounter rate with sparse, randomly distributed, replenishable targets when the search restarts in the vicinity of the previously visited target, which becomes revisitable again with high probability, i.e., non-destructive foraging [Nature 401, 911 (1999)]. The precise conditions for the validity of this Lévy flight foraging hypothesis (LFH) have been widely described in the literature [Phys. Life Rev. 14, 94 (2015)]. Nevertheless, three objecting claims to the LFH have been raised recently for $d \geq 2$: (i) the capture rate $η$ has linear dependence on the target density $ρ$ for all values of the Lévy index $α$; (ii) "the gain $η_{max}/η$ achieved by varying $α$ is bounded even in the limit $ρ\to 0 $" so that "tuning $α$ can only yield a marginal gain"; (iii) depending on the values of the radius of detection $a$, the restarting distance $l_c$ and the scale parameter $s$, the optimum is realized for a range of $α$ [Phys. Rev. Lett. 124, 080601 (2020)]. Here we answer each of these three criticisms in detail and show that claims (i)-(iii) do not actually invalidate the LFH. Our results and analyses restore the original result of the LFH for non-destructive foraging.

cond-mat.stat-mech↗

arXiv2020

The connection between Jackson and Hausdorff derivatives in the context of generalized statistical mechanics

In literature one can find many generalizations of the usual Leibniz derivative, such as Jackson derivative, Tsallis derivative and Hausdorff derivative. In this article we present a connection between Jackson derivative and recently proposed Hausdorff derivative. On one hand, the Hausdorff derivative has been previously associated with non-extensivity in systems presenting fractal aspects. On the other hand, the Jackson derivative has a solid mathematical basis because it is the $\overline{q}$-analog of the ordinary derivative and it also arises in quantum calculus. From a quantum deformed $\overline{q}$-algebra we obtain the Jackson derivative and then address the problem of $N$ non-interacting quantum oscillators. We perform an expansion in the quantum grand partition function from which we obtain a relationship between the parameter $\overline{q}$, related to Jackson derivative, and the parameters $ζ$ and $q$ related to Hausdorff derivative and Tsallis derivative, respectively.

cond-mat.stat-mech↗

arXiv2020

Correspondence between spanning trees and the Ising model on a square lattice

An important problem in statistical physics concerns the fascinating connections between partition functions of lattice models studied in equilibrium statistical mechanics on the one hand and graph theoretical enumeration problems on the other hand. We investigate the nature of the relationship between the number of spanning trees and the partition function of the Ising model on the square lattice. The spanning tree generating function $T(z)$ gives the spanning tree constant when evaluated at $z=1$, while giving he lattice green function when differentiated. It is known that for the infinite square lattice the partition function $Z(K)$ of the Ising model evaluated at the critical temperature $K=K_c$ is related to $T(1)$. Here we show that this idea in fact generalizes to all real temperatures. We prove that $ ( Z(K) {\rm sech~} 2K ~\!)^2 = k \exp\big[ T(k) \big] $, where $k= 2 \tanh(2K) {\rm sech}(2K)$. The identical Mahler measure connects the two seemingly disparate quantities $T(z)$ and $Z(K)$. In turn, the Mahler measure is determined by the random walk structure function. Finally, we show that the the above correspondence does not generalize in a straightforward manner to non-planar lattices.

cond-mat.stat-mech↗

arXiv2018

Characterizing complex networks using Entropy-degree diagrams: unveiling changes in functional brain connectivity induced by Ayahuasca

Open problems abound in the theory of complex networks, which has found successful application to diverse fields of science. With the aim of further advancing the understanding of the brain's functional connectivity, we propose to evaluate a network metric which we term the geodesic entropy. This entropy, in a way that can be made precise, quantifies the Shannon entropy of the distance distribution to a specific node from all other nodes. Measurements of geodesic entropy allow for the characterization of the structural information of a network that takes into account the distinct role of each node into the network topology. The measurement and characterization of this structural information has the potential to greatly improve our understanding of sustained activity and other emergent behaviors in networks, such as self-organized criticality sometimes seen in such contexts. We apply these concepts and methods to study the effects of how the psychedelic Ayahuasca affects the functional connectivity of the human brain. We show that the geodesic entropy is able to differentiate the functional networks of the human brain in two different states of consciousness in the resting state: (i) the ordinary waking state and (ii) a state altered by ingestion of the Ayahuasca. The entropy of the nodes of brain networks from subjects under the influence of Ayahuasca diverge significantly from those of the ordinary waking state. The functional brain networks from subjects in the altered state have, on average, a larger geodesic entropy compared to the ordinary state. We conclude that geodesic entropy is a useful tool for analyzing complex networks and discuss how and why it may bring even further valuable insights into the study of the human brain and other empirical networks.

q-bio.NC↗

arXiv2017

The complex social network of surnames: A comparison between Brazil and Portugal

We present a study of social networks based on the analysis of Brazilian and Portuguese family names (surnames). We construct networks whose nodes are names of families and whose edges represent parental relations between two families. From these networks we extract the connectivity distribution, clustering coefficient, shortest path and centrality. We find that the connectivity distribution follows an approximate power law. We associate the number of hubs, centrality and entropy to the degree of miscegenation in the societies in both countries. Our results show that Portuguese society has a higher miscegenation degree than Brazilian society. All networks analyzed lead to approximate inverse square power laws in the degree distribution. We conclude that the thermodynamic limit is reached for small networks (3 or 4 thousand nodes). The assortative mixing of all networks is negative, showing that the more connected vertices are connected to vertices with lower connectivity. Finally, the network of surnames presents some small world characteristics.

physics.soc-ph↗

arXiv2016

Shannon entropy of brain functional complex networks under the influence of the psychedelic Ayahuasca

The entropic brain hypothesis holds that the key facts concerning psychedelics are partially explained in terms of increased entropy of the brain's functional connectivity. Ayahuasca is a psychedelic beverage of Amazonian indigenous origin with legal status in Brazil in religious and scientific settings. In this context, we use tools and concepts from the theory of complex networks to analyze resting state fMRI data of the brains of human subjects under two distinct conditions: (i) under ordinary waking state and (ii) in an altered state of consciousness induced by ingestion of Ayahuasca. We report an increase in the Shannon entropy of the degree distribution of the networks subsequent to Ayahuasca ingestion. We also find increased local and decreased global network integration. Our results are broadly consistent with the entropic brain hypothesis. Finally, we discuss our findings in the context of descriptions of "mind-expansion" frequently seen in self-reports of users of psychedelic drugs.

q-bio.NC↗

arXiv2015

Information entropy of classical versus explosive percolation

We study the Shannon entropy of the cluster size distribution in classical as well as explosive percolation, in order to estimate the uncertainty in the sizes of randomly chosen clusters. At the critical point the cluster size distribution is a power-law, i.e. there are clusters of all sizes, so one expects the information entropy to attain a maximum. As expected, our results show that the entropy attains a maximum at this point for classical percolation. Surprisingly, for explosive percolation the maximum entropy does not match the critical point. Moreover, we show that it is possible determine the critical point without using the conventional order parameter, just analysing the entropy's derivatives.

cond-mat.dis-nn↗

arXiv2015

The hypergeometric series for the partition function of the 2-D Ising model

In 1944 Onsager published the formula for the partition function of the Ising model for the infinite square lattice. He was able to express the internal energy in terms of a special function, but he left the free energy as a definite integral. Seven decades later, the partition function and free energy have yet to be written in closed form, even with the aid of special functions. Here we evaluate the definite integral explicitly, using hypergeometric series. Let $β$ denote the reciprocal temperature, $J$ the coupling and $f$ the free energy per spin. We prove that $-βf = \ln(2 \cosh 2K) - κ^2\, {}_4F_3 [1,1,\tfrac{3}{2},\tfrac{3}{2};\ 2,2,2 ;\ 16 κ^2 ] $, where $_p F_q$ is the generalized hypergeometric function, $K=βJ$, and $2κ= {\rm tanh} 2K {\rm sech} 2K$.

cond-mat.stat-mech↗